[EM] Notes on a few Later-no-harm methods

[Richard Lung]

Hello Kevin,

Avoiding divisions by zero is a different unrelated issue.
It applies to the geometric mean, the most essential average for binomial stv.
No, it's just that the binomial distribution doesn't become apparent, except with larger numbers of items.
The omission of a candidate preference implies an abstention, which counts towards a vacancy quota. All preferences are counted: conservation of information. (It led to the holographic principle, in physics!)

Tossing a coin just means that, for voters with no preference between 2 candidates, they will leave it to chance, who gets the prior preference, so that neither one will get an over-all undeserved advantage, to any statistically significant degree. This does assume a large enough sample to iron out any chance advantage to one of the unprefered candidates.

Nothing of the sort, you conjecture, takes place, in binomial stv, which does not omit candidates. That is the point of the exclusion count. BSTV is system of keep values: the quota divided by candidates votes, in surplus or deficit of a quota; counting preferences and reverse preferences (election and exclusion counts).


Regards,
Richard Lung.



On 14 May 2022, at 5:36 pm, Kevin Venzke <stepjak at yahoo.fr> wrote:

Hi Richard,

> Binomial stv is a statistical count that doesn't apply for very small
> numbers. For that, there is non-parametric statistics. There is no hard
> and fast rule. I'd say about 32 votes minimum. but that's just a
> minimum. There is a law of large numbers for better approximations.

Do you say it doesn't apply to very small numbers because you seek to avoid
divisions by zero in the math? I don't think there is any minimum number of voters
that will guarantee that.

> I forget the meaning of truncated, kindly explained to me.

It's the omission of a candidate from a voter's ranking.

> If a voter is really stuck between two candidates, he or she can toss a
> coin for it, and there will be no over-all bias.

This does bring to mind to a suggestion which would *not* satisfy Later-no-harm:
Suppose that BSTV automatically splits the voter into two when they omit the last
two candidates:

480: A      <-- These voters omit B and C
405: B>C>A
115: C>A>B

So BSTV would automatically split them into two blocs to achieve the following
ballots:

240: A>B>C  <-- half of the bloc
240: A>C>B  <-- the other half
405: B>C>A
115: C>A>B

BSTV elects A.

But now, suppose that the 480 voters had actually decided that they felt that B is
better than C, so they instead voted like this:

480: A>B>C  <-- different vote
405: B>C>A
115: C>A>B

Now BSTV elects B. The A>B>C voters do not prefer that outcome, so they were
better off omitting the lower rankings.

That can never happen in ordinary STV, so we say STV satisfies Later-no-harm.

Kevin